We also note a strong correlation between different mixing times, see Figure REF .
In particular, the mixing times in the quadrangulation and its dual are increasingly correlated as the size of the quadrangulation increases; we draw a parallel with [1]}, where the authors prove that triangulations and their duals are asymptotically isometric in the large scale as their size goes to \(+\infty \)  (the results can be generalized to quadrangulations, as is done in [2]}, although it does not handle the dual). This leads us to conjecture that it is the macroscopic scale, rather than the microscopic scale, that influences the mixing time the most, in the sense that if two maps are asymptotically isometric when their distances are appropriately rescaled, then their mixing times will be asymptotically equal.
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